Whether you can do so is a question that only you can answer, and you can answer it by trying to build one. A web search for "homemade cyclotron" yields many interesting articles, videos and sites. Whether those techniques are within your level of skill and budget, how should we know?
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Eric LippertJul 8 '14 at 16:09

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In the general sense all you need is a source of particles and an accelerating field. Both exist in a fluorescent tube or a CRT. With a Van der Graaff generator you can make quite high voltages - 10's of kV - with which to accelerate electrons / make "lightning". But even a BB gun is a "particle" accelerator. You might have to be more specific.
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FlorisJul 8 '14 at 19:25

3 Answers
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It is not so hard, but it won't be able to generate enough high energetic particles.

The best example for a particle accelerator is a CRT (cathod ray tube), which you can find in every CRT monitor or TV. It can generate around $40keV$ electrons. (LHC generates $3.5TeV$ protons, thus it is around a hundred million times stronger).

Only a particle accelerator isn't enough, if you want to make experiments with it, you need some analytic/measurements devices also. In the current accelerators, they are nearly so complex and costly as the main accelerating device.

Maybe it is not a particle accelerator in the classical sense, it creates enough strong field to be able to fuse deuterons (although it is doing this with terrible efficiency, around $10^{-8}$). You can see a Farnsworth fusor scematic below:

Besides the already suggested cathode ray tube, another rather small accelerator would be a cyclotron. The first one was built by Lawrence and Livingston, "a device about 4.5 inches in diameter used a potential of 1,800 volts to accelerate hydrogen ions up to energies of 80,000 electron volts." I guess it would be possible to get a high voltage power source and hydrogen, but really building a working device would probably take a long time.

The ionization energy of nuclear hydrogen is 13,6eV, which corresponds to electromagnetic radiation of 91nm wavelength (extreme UV). So you could use a "light" source emitting a wavelength smaller than 91nm and just direct it at the hydrogen. Another way would be to heat it up. However, there might be easier solutions, e.g. by applying an electric field, but i'm not sure about this. Interesting question, though.
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TimitryJul 8 '14 at 15:15

Another idea: Maybe you could use a thin stripper foil (usually made from carbon) and direct your gas with some pressure through it. This is how negative ions are converted into positive ones in e.g. a Tandem Van de Graaff accelerator. However, i don't know if it will work with "slow" particles from a gas bottle.
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TimitryJul 9 '14 at 7:57

The laser plasma accelerator has accelerated about half a billion
electrons to 2 gigaelectronvolts over a distance of about 1 inch. It's
a downsizing of a factor of approximately 10,000, and marks a major
milestone in the advance toward the day when multi-gigaelectronvolt
laser plasma accelerators are standard equipment in research
laboratories around the world.